Two Kinds of Concept Introduction Paul M. Pietroski University of Maryland Dept. of Linguistics, Dept. of Philosophy

Human Language System, tuned to Spoken English Sound(Brutus kicked Caesar) Meaning(Brutus kicked Caesar) B RUTUS KICKED (_, _) C AESAR KICKED (B RUTUS, C AESAR ) Human Language System, tuned to Spoken English Sound(kicked) Meaning(kicked) KICKED (_, C AESAR ) KICKED (_, _) Y. X. KICKED ( X, Y )

Human Language System, tuned to Spoken English Sound(Brutus kicked Caesar on Monday) Meaning(Brutus kicked Caesar on Monday) B RUTUS KICKED (_, _, _) C AESAR KICKED (_, B RUTUS, C AESAR ) & ON (_, M ONDAY ) ON (_, M ONDAY ) KICKED (_, _, C AESAR ) KICKED (_, B RUTUS, C AESAR )

Human Language System, tuned to Spoken English Sound(kicked) Meaning(kicked) B RUTUS KICKED (_, _, _) C AESAR ON (_, M ONDAY ) KICKED (_, B RUTUS, C AESAR ) Y. X. E. KICKED ( E, X, Y )

Human Language System, tuned to Spoken English Sound(kicked) Meaning(kicked) B RUTUS KICKED (_, _, _) C AESAR ON (_, M ONDAY ) KICKED (_, B RUTUS, C AESAR ) Y. X. E. KICK ( E, X, Y ) & PAST ( E )

Human Language System, tuned to Spoken English Sound(kicked) Meaning(kicked) B RUTUS KICKED (_, _, _) C AESAR ON (_, M ONDAY ) KICKED (_, B RUTUS, C AESAR ) Y. X. E. KICK ( E, X, Y ) & PAST ( E ) & AGENT ( E, X ) & PATIENT ( E, Y )

Human Language System, tuned to Spoken English Sound(kicked) Meaning(kicked) Brutus kicked Caesar (today) Brutus kicked Brutus kicked Caesar the ball Caesar was kicked I get no kick from champagne, but I get a kick out of you That kick was a good one Y. X. E. KICKED ( E, X, Y ) Does a child who acquires kicked start with KICKED (_, _) Y. X. KICKED ( X, Y ) KICK (_, _, _) Y. X. E. KICK ( E, X, Y ) or some other concept(s), perhaps like E. KICK ( E )?

Human Language System, tuned to Spoken English Sound(gave) Meaning(gave) Brutus gave the ball (to Caesar) Brutus gave (at the office) Brutus gave Caesar the ball (today) The painting was given/donated Z. Y. X. E. GAVE ( E, X, Y, Z ) Does a child who acquires gave start with GAVE (_, _, _) GAVE (_, _, _, _) Z. Y. X. E. GAVE ( E, X, Y, Z ) … E. GIVE ( E ) The rope has too much give

Big Questions To what degree is lexical acquisition a cognitively passive process in which (antecedently available) representations are simply labeled and paired with phonological forms? To what degree is lexical acquisition a cognitively creative process in which (antecedently available) representations are used to introduce other representations that exhibit a new combinatorial format? Are the concepts that kids lexicalize already as systematically combinable as words? Or do kids use these L-concepts, perhaps shared with other animals, to introduce related concepts that exhibit a more distinctively human format?

Outline Describe a Fregean mind that can use its cognitive resources to introduce mental symbols of two sorts (1) polyadic concepts that are logically fruitful (2) monadic concepts that are logically boring, but more useful than mere abbreviations like X [ MARE ( X ) DF HORSE ( X ) & FEMALE ( X ) & MATURE ( X )] Suggest that the process of acquiring a lexicon involves concept introduction of the second sort. For example, a child might use GIVE (_, _, _) or GIVE (_, _, _, _) to introduce GIVE (_). Offer some evidence that this suggestion is correct say something about adicity

Quick Reminder about Conceptual Adicity Two Common Metaphors Jigsaw Puzzles 7 th Grade Chemistry H–O–H +1

Jigsaw Metaphor a Thought: B RUTUS SANG a Thought: B RUTUS SANG

Jigsaw Metaphor Unsaturated Saturater Doubly Un- saturated 1st saturater 2nd saturater one Monadic Concept (adicity: -1) filled by one Saturater (adicity +1) yields a complete Thought one Dyadic Concept (adicity: -2) filled by two Saturaters (adicity +1) yields a complete Thought B RUTUS S ANG ( ) B RUTUS C AESAR KICK(_, _)

7 th Grade Chemistry Metaphor a molecule of water a molecule of water H(OH +1 ) -1 a single atom with valence -2 can combine with two atoms of valence +1 to form a stable molecule

7 th Grade Chemistry Metaphor Brutus(KickCaesar +1 ) -1

7 th Grade Chemistry Metaphor +1 NaCl -1 an atom with valence -1 can combine with an atom of valence +1 to form a stable molecule +1 BrutusSang -1

Extending the Metaphor A GGIE BROWN ( ) A GGIE COW ( ) A GGIE BROWN - COW ( ) Aggie is (a) cow Aggie is brown Aggie is (a) brown cow +1

Extending the Metaphor A GGIE Conjoining two monadic (-1) concepts can yield a complex monadic (-1) concept BROWN ( ) & COW ( ) A GGIE COW ( ) +1 A GGIE BROWN ( ) +1

Outline Describe a Fregean mind that can use its cognitive resources to introduce mental symbols of two sorts (1) polyadic concepts that are logically fruitful (2) monadic concepts that are logically boring, but more useful than mere abbreviations like X [ MARE ( X ) DF HORSE ( X ) & FEMALE ( X ) & MATURE ( X )] Suggest that for humans, the process of acquiring a lexicon involves concept introduction of the second sort. For example, a child might use GIVE (_, _, _) to introduce GIVE (_). Offer some evidence that this suggestion is correct

Zero is a number every number has a successor Zero is not a successor of any number no two numbers have the same successor for every property of Zero: if every number that has it is such that its successor has it, then every number has it Freges Hunch: these thoughts are not as independent as axioms should be

Nero is a cucumber every cucumber has a doubter Nero is not a doubter of any cucumber no two cucumbers have the same doubter for every property of Nero: if every cucumber that has it is such that its doubter has it, then every cucumber has it Related Point: these thoughts are not as compelling as axioms should be

Zero is a number ( ) Nero is a cucumber every number has a successor x{ (x) y[ (y,x)]} every cucumber has a doubter Zero is not a successor of any number ~ x{ (, x) & (x)} Nero is not a doubter of any cucumber If we want to represent what the arithmetic thougts imply and what they follow fromthen the formalizations on the right are better than P, Q, and R.

Zero is a number ( ) Nero is a cucumber every number has a successor x{ (x) y[ (y,x)]} every cucumber has a doubter Zero is not a successor of any number ~ x{ (, x) & (x)} Nero is not a doubter of any cucumber But we miss something if we represent the property of being a number with an atomic monadic concept, as if NUMBER ( ) differs from CUCUMBER ( ) and CARROT ( ) only in terms of its specific content.

Zero is a number ( ) Nero is a cucumber every number has a successor x{ (x) y[ (y,x)]} every cucumber has a doubter Zero is not a successor of any number ~ x{ (, x) & (x)} Nero is not a doubter of any cucumber We miss something if we represent the number zero with an atomic singular concept, as if the concept Z ERO differs from concepts like N ERO and V ENUS only in terms of its specific content.

Zero is a number ( ) Nero is a cucumber every number has a successor x{ (x) y[ (y,x)]} every cucumber has a doubter Zero is not a successor of any number ~ x{ (, x) & (x)} Nero is not a doubter of any cucumber We miss something if we represent the successor relation with an atomic relational concept, as if SUCCESSOR - OF (, ) differs from DOUBTER - OF (, ) and PLANET - OF (, ) only in terms of its specific content.

Fregean Moral The contents of NUMBER ( ), Z ERO, and SUCCESSOR - OF (, ) seem to be logically related in ways that the contents of CUCUMBER ( ), N ERO, and DOUBTER - OF (, ) are not So perhaps we should try to re-present the contents of the arithmetic concepts, and reduce (re-presentations of) the arithmetic axioms to sparer axioms that reflect (all and only) the nonlogical content of arithmetic

Freges Success ( with help from Wright/Boolos/Heck ) (Defs) Definitions of NUMBER (x), Z ERO, and SUCCESSOR - OF (y, x) in terms of NUMBER - OF (x, ) and logical notions (HP) {[ x: NUMBER - OF (x, ) = NUMBER - OF ( X, )] ONE - TO - ONE (, )} __________________________________________ (DP) Dedekind-Peano Axioms A mind that starts with (HP) could generate Frege Arithmetic, given a consistent fragment of Freges (2 nd -order) Logic.

Some Truths NUMBER - OF [Z ERO, z.~(z = z)] NUMBER - OF [O NE, z.(z = Z ERO )] NUMBER - OF [T WO, z.(z = Z ERO ) v (z = O NE )]... Z ERO = x: NUMBER - OF [x, z.~(z = z)] O NE = x: NUMBER - OF [x, z.(z = Z ERO )] T WO = x: NUMBER - OF [x, z.(z = Z ERO ) v (z = O NE )] … x { NUMBER - OF (x, ) NUMBER (x) & ONE - TO - ONE (, λy. PRECEDES (y, x)]} x{ NUMBER (x) (x = Z ERO ) v PRECEDES (Z ERO, x)} x{ NUMBER (x) [ NUMBER - OF (x, )]} some initial clues to let us cotton on and get the concept NUMBER - OF [x, ] We can call these definitions in the OED sense without saying that the truths are analytic.

Freges Direction of Definition Z ERO = DF x: NUMBER - OF [x, z.~(z = z)] x{ NUMBER (x) DF (x = Z ERO ) v PRECEDES (Z ERO, x)} x y{ PRECEDES (x, y) DF ANCESTRAL: PREDECESSOR (x, y)} x y{ PREDECESSOR (x, y) DF { NUMBER - OF (x, ) & NUMBER - OF (y, ) & z[~ z & w{ w [ w v (w = z)]}]} The idea is not that our concepts Z ERO, NUMBER ( ), and PREDECESSOR (x, y) had decompositions all along. But once we have the concept NUMBER - OF (x, ), perhaps with help from Frege, we can imagine an ideal thinker who starts with this concept and introduces the otherseither to interpret us, or to suppress many aspects of logical structure until they become relevant.

A Different Direction of Analysis x { NUMBER - OF (x, ) DF NUMBER (x) & ONE - TO - ONE (, λy. PRECEDES (y, x)]} x y{ PRECEDES (x, y) DF ANCESTRAL: PREDECESSOR (x, y)} x y[ PREDECESSOR (x, y) DF SUCCESSOR (y, x)] as if primitive thinkerswho start with NUMBER (x), ONE - TO - ONE (, ) and SUCCESSOR (x, y)were trying to interpret (or become) thinkers who use the sophisticated concept NUMBER - OF (x, ) and thereby come to see how the Dedekind-Peano axioms can be re-presented such thinkers dont use NUMBER - OF (x, ) to introduce NUMBER (x) but they might use KICK (x, y)/ GIVE (x, y, x) to introduce KICK (e)/ GIVE (e)

Some Potential Equivalences x{ SOCRATIZES (x) (x = S OCRATES )} S OCRATES = x: SOCRATIZES (x) x y{ KICKED (x, y) e[ KICKED (e, x, y)]} x y e{ KICKED (e, x, y) PAST (e) & KICK (e, x, y)} x y e{ KICK (e, x, y) AGENT (e, x) & KICK (e) & PATIENT (e, y)} λe. KICK (e) = : x y e{ KICK (e, x, y) AGENT (e, x) & (e) & PATIENT (e, y)}

Some Potential Equivalences x{ SOCRATIZES (x) (x = S OCRATES )} fixes the (one-element) extension of SOCRATIZES (_) x y e{ KICK (e, x, y) AGENT (e, x) & KICK (e) & PATIENT (e, y)} λe. KICK (e) = : x y e{ KICK (e, x, y) AGENT (e, x) & (e) & PATIENT (e, y)} constrains (but does not fix) the extension of KICK (_), which is neither fully defined nor an unconstrained atom

Two Kinds of Introduction Introduce monadic concepts in terms of nonmonadic concepts x{ SOCRATIZES (x) DF (x = S OCRATES ) λe. KICK (e) = DF : x y e{ KICK (e, x, y) AGENT (e, x) & (e) & PATIENT (e, y)} _____________________________________________________ Introduce nonmonadic concepts in terms of monadic concepts S OCRATES = DF x: SOCRATIZES (x) x y e{ KICK (e, x, y) DF AGENT (e, x) & KICK (e) & PATIENT (e, y)}

Two Kinds of Introduction Introduce monadic concepts in terms of nonmonadic concepts x{ SOCRATIZES (x) DF (x = S OCRATES ) λe. KICK (e) = DF : x y e{ KICK (e, x, y) AGENT (e, x) & (e) & PATIENT (e, y)} _____________________________________________________ My suggestion is not that our concepts S OCRATES and KICK (e, x, y) have decompositions. But a child who starts with these concepts might introduce KICK (e) and SOCRATIZES (x).

Two Kinds of Introduction Introduce monadic concepts in terms of nonmonadic concepts x{ SOCRATIZES (x) DF (x = S OCRATES ) λe. KICK (e) = DF : x y e{ KICK (e, x, y) AGENT (e, x) & (e) & PATIENT (e, y)} _____________________________________________________ The monadic conceptsunlike NUMBER - OF (x, )wont help much if your goal is to show how logic is related to arithmetic. But they might help if you want to specify word meanings in a way that allows for logical forms that involve using conjunction to build complex monadic predicates as in… e{ x[ AGENT (e, x) & SOCRATIZES (x)] & KICK (e) & PAST (e)}

Two Conceptions of Lexicalization (1) Labelling Acquiring words is basically a process of pairing pre-existing concepts with perceptible signals Lexicalization is a conceptually passive operation Word combination mirrors concept combination (2) Reformatting Acquiring words is also a process of introducing concepts that exhibit a certain composition format In this sense, lexicalization is cognitively creative Word combination does not mirror combination of the pre-existing concepts that get lexicalized

Bloom: How Children Learn the Meanings of Words word meanings are, at least primarily, concepts that kids have prior to lexicalization learning word meanings is, at least primarily, a process of figuring out which existing concepts are paired with which word-sized signals in this process, kids draw on many capacitiese.g., recognition of syntactic cues and speaker intentions but no capacities specific to acquiring word meanings

Lidz, Gleitman, and Gleitman Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence is a function of the number of participants logically implied by the verb meaning. It takes only one to sneeze, and therefore sneeze is intransitive, but it takes two for a kicking act (kicker and kickee), and hence kick is transitive. Of course there are quirks and provisos to these systematic form-to-meaning-correspondences… Brutus kicked Caesar the ball That kick was a good one Brutus kicked Caesar was kicked

Lidz, Gleitman, and Gleitman Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence is a function of the number of participants logically implied by the verb meaning. It takes only one to sneeze, and therefore sneeze is intransitive, but it takes two for a kicking act (kicker and kickee), and hence kick is transitive. Of course there are quirks and provisos to these systematic form-to-meaning-correspondences…

Why Not... Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence is not a function of the number of participants logically implied by the verb meaning. A paradigmatic act of kicking has exactly two participants (kicker and kickee), and yet kick need not be transitive. Brutus kicked Caesar the ball Caesar was kicked Brutus kicked Brutus gave Caesar a swift kick Of course there are quirks and provisos. Some verbs do require a certain number of noun phrases in active voice sentences. *Brutus put the ball *Brutus put *Brutus sneezed Caesar

Concept of adicity n Concept of adicity n Concept of adicity n Concept of adicity n Perceptible Signal Quirky information for lexical items like kick Concept of adicity -1 Concept of adicity -1 Perceptible Signal Quirky information for lexical items like put

Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence is a function of the number of participants logically implied by the verb meaning. It takes only one to sneeze, and therefore sneeze is intransitive, but it takes two for a kicking act (kicker and kickee), and hence kick is transitive. Of course there are quirks and provisos to these systematic form-to-meaning-correspondences. Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence isnt a function of the number of participants logically implied by the verb meaning. It takes only one to sneeze, and usually sneeze is intransitive. But it usually takes two to have a kicking; and yet kick can be untransitive. Of course there are quirks and provisos. Some verbs do require a certain number of noun phrases in active voice sentences.

Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence is a function of the number of participants logically implied by the verb meaning. It takes only one to sneeze, and therefore sneeze is intransitive, but it takes two for a kicking act (kicker and kickee), and hence kick is transitive. Of course there are quirks and provisos to these systematic form-to-meaning-correspondences. Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence isnt a function of the number of participants logically implied by the verb meaning. It takes only one to sneeze, and sneeze is typically used intransitively; but a paradigmatic kicking has exactly two participants, and yet kick can be used intransitively or ditransitively. Of course there are quirks and provisos. Some verbs do require a certain number of noun phrases in active voice sentences.

Lexicalization as Concept-Introduction (not mere labeling) Concept of type T Concept of type T Concept of type T Concept of type T Concept of type T* Concept of type T* Perceptible Signal

Lexicalization as Concept-Introduction (not mere labeling) Perceptible Signal Number(_) type: Number(_) type: Number(_) type: Number(_) type: NumberOf[_, Φ(_)] type:, > NumberOf[_, Φ(_)] type:, >

Lexicalization as Concept-Introduction (not mere labeling) Concept of type T Concept of type T Concept of type T Concept of type T Concept of type T* Concept of type T* Perceptible Signal

Concept of adicity -1 Concept of adicity -1 Concept of adicity -1 Concept of adicity -1 Concept of adicity -2 Concept of adicity -2 Perceptible Signal ARRIVE(x) ARRIVE(e, x) One Possible (Davidsonian) Application: Increase Adicity

Concept of adicity -2 Concept of adicity -2 Concept of adicity -2 Concept of adicity -2 Concept of adicity -3 Concept of adicity -3 Perceptible Signal KICK(x 1, x 2 ) KICK(e, x 1, x 2 ) One Possible (Davidsonian) Application: Increase Adicity

Concept of adicity n Concept of adicity n Concept of adicity n Concept of adicity n Concept of adicity -1 Concept of adicity -1 Perceptible Signal KICK(x 1, x 2 ) KICK(e) KICK(e, x 1, x 2 ) Lexicalization as Concept-Introduction: Make Monads

Concept of adicity n Concept of adicity n Concept of adicity n (or n1) Concept of adicity n (or n1) Perceptible Signal Concept of adicity n Concept of adicity n Concept of adicity 1 Concept of adicity 1 Perceptible Signal Further lexical information (regarding flexibilities) further lexical information (regarding inflexibilities) Two Pictures of Lexicalization

Concept of adicity n Concept of adicity n Concept of adicity n Concept of adicity n Concept of adicity 1 Concept of adicity 1 Perceptible Signal further POSSE information, as for put Two Pictures of Lexicalization Word: SCAN -1 offer some reminders of some reasons (in addition to passives/nominalizations) for adopting the second picture

Absent Word Meanings Striking absence of certain (open-class) lexical meanings that would be permitted if Human I -Languages permitted nonmonadic semantic types <e,<e,<e,<e, t>>>> (instructions to fetch) tetradic concepts <e,<e,<e, t>>> (instructions to fetch) triadic concepts <e,<e, t>> (instructions to fetch) dyadic concepts <e> (instructions to fetch) singular concepts

Striking absence of certain (open-class) lexical meanings that would be permitted if I-Languages permit nonmonadic semantic types >>> (instructions to fetch) tetradic concepts >> (instructions to fetch) triadic concepts > (instructions to fetch) dyadic concepts (instructions to fetch) singular concepts

Absent Word Meanings Brutus sald a car Caesar a dollar sald SOLD(x, $, z, y) [sald [a car]] SOLD(x, $, z, a car) [[sald [a car]] Caesar] SOLD(x, $, Caesar, a car) [[[sald [a car]] Caesar]] a dollar] SOLD(x, a dollar, Caesar, a car) _________________________________________________ Caesar bought a car bought a car from Brutus for a dollar bought Antony a car from Brutus for a dollar x sold y to z (in exchange) for $

Absent Word Meanings Brutus tweens Caesar Antony tweens BETWEEN(x, z, y) [tweens Caesar] BETWEEN(x, z, Caesar) [[tweens Caesar] Antony] BETWEEN(x, Antony, Caesar) _______________________________________________________ Brutus sold Caesar a car Brutus gave Caesar a car*Brutus donated a charity a car Brutus gave a car away Brutus donated a car Brutus gave at the office Brutus donated anonymously

Absent Word Meanings Alexander jimmed the lock a knife jimmed JIMMIED(x, z, y) [jimmed [the lock] JIMMIED(x, z, the lock) [[jimmed [the lock] [a knife]] JIMMIED(x, a knife, the lock) _________________________________________________ Brutus froms Rome froms COMES-FROM(x, y) [froms Rome] COMES-FROM(x, Rome)

Absent Word Meanings Brutus talls Caesar talls IS-TALLER-THAN(x, y) [talls Caesar] IS-TALLER-THAN(x, Caesar) _________________________________________ *Julius Caesar Julius JULIUS Caesar CAESAR

Absent Word Meanings Striking absence of certain (open-class) lexical meanings that would be permitted if I-Languages permit nonmonadic semantic types >>> (instructions to fetch) tetradic concepts >> (instructions to fetch) triadic concepts > (instructions to fetch) dyadic concepts (instructions to fetch) singular concepts

Proper Nouns even English tells against the idea that lexical proper nouns label singular concepts (of type ) Every Tyler I saw was a philosopher Every philosopher I saw was a Tyler There were three Tylers at the party That Tyler stayed late, and so did this one Philosophers have wheels, and Tylers have stripes The Tylers are coming to dinner I spotted Tyler Burge I spotted that nice Professor Burge who we met before proper nouns seem to fetch monadic concepts, even if they lexicalize singular concepts

Concept of adicity n Concept of adicity n Concept of adicity n Concept of adicity n Concept of adicity -1 Concept of adicity -1 Perceptible Signal TYLER TYLER(x) CALLED[x, SOUND(Tyler)] Lexicalization as Concept-Introduction: Make Monads

Concept of adicity n Concept of adicity n Concept of adicity n Concept of adicity n Concept of adicity -1 Concept of adicity -1 Perceptible Signal KICK(x 1, x 2 ) KICK(e) KICK(e, x 1, x 2 ) Lexicalization as Concept-Introduction: Make Monads

Concept of adicity n Concept of adicity n Concept of adicity n Concept of adicity n Concept of adicity -1 Concept of adicity -1 Perceptible Signal TYLER TYLER(x) CALLED[x, SOUND(Tyler)] Lexicalization as Concept-Introduction: Make Monads

Lexical Idiosyncracy can be Lexically Encoded A verb can access a monadic concept and impose further (idiosyncratic) restrictions on complex expressions Semantic Composition Adicity Number (SCAN) (instructions to fetch) singular concepts +1 singular (instructions to fetch) monadic concepts -1 monadic (instructions to fetch) dyadic concepts -2 dyadic > Property of Smallest Sentential Entourage (POSSE) zero NPs, one NP, two NPs, … the SCAN of every verb can be -1, while POSSEs vary: zero, one, two, …

POSSE facts may reflect...the adicities of the original concepts lexicalized...statistics about how verbs are used (e.g., in active voice)...prototypicality effects...other agrammatical factors put may have a (lexically represented) POSSE of three in part because --the concept lexicalized was PUT(_, _, _) --the frequency of locatives (as in put the cup on the table) is salient and note: * I put the cup the table ? I placed the cup

On any view: Two Kinds of Facts to Accommodate Flexibilities – Brutus kicked Caesar – Caesar was kicked – The baby kicked – I get a kick out of you – Brutus kicked Caesar the ball Inflexibilities – Brutus put the ball on the table – *Brutus put the ball – *Brutus put on the table

On any view: Two Kinds of Facts to Accommodate Flexibilities – The coin melted – The jeweler melted the coin – The fire melted the coin – The coin vanished – The magician vanished the coin Inflexibilities – Brutus arrived – *Brutus arrived Caesar

Experience and Growth Language Acquisition Device in its Initial State Language Acquisition Device in a Mature State (an I-Language): GRAMMAR LEXICON Phonological Instructions Semantic Instructions Lexicalizable concepts Introduced concepts Articulation and Perception of Signals Lexicalized concepts

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